To calculate the work done in compressing the gas isothermally, we can use the formula for isothermal work done on an ideal gas:
\[ W = m \cdot R \cdot T \cdot \ln\left(\frac{P_2}{P_1}\right) \]
Where:
- \( W \) = work done
- \( m \) = mass of the gas (in kg)
- \( R \) = specific gas constant (in kJ/kg·K)
- \( T \) = absolute temperature (in K)
- \( P_1 \) = initial pressure (in kPa)
- \( P_2 \) = final pressure (in kPa)
- \( \ln \) = natural logarithm
Given data:
- \( m = 1.36 \) kg
- \( R = 0.286 \) kJ/kg·K
- \( T = 77 \) °C = \( 77 + 273.15 = 350.15 \) K
- \( P_1 = 138 \) kPa
- \( P_2 = 1380 \) kPa
Calculating the Work Done:
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Calculate the Natural Logarithm of the Pressure Ratio: \[ \frac{P_2}{P_1} = \frac{1380 \text{ kPa}}{138 \text{ kPa}} = 10 \] \[ \ln\left(\frac{P_2}{P_1}\right) = \ln(10) \approx 2.3026 \]
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Substitute the Values into the Work Done Formula: \[ W = 1.36 , \text{kg} \cdot 0.286 , \text{kJ/kg·K} \cdot 350.15 , \text{K} \cdot 2.3026 \]
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Calculate the Result Step-by-Step:
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First calculate \( m \cdot R \cdot T \): \[ W = 1.36 \cdot 0.286 \cdot 350.15 \cdot 2.3026 \]
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Calculate \( m \cdot R \cdot T \): \[ 1.36 \cdot 0.286 = 0.38896 , \text{kJ/K} \] \[ 0.38896 \cdot 350.15 \approx 136.286 , \text{kJ} \]
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Finally multiply by the logarithm term: \[ W = 136.286 \cdot 2.3026 \approx 313.385 , \text{kJ} \]
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Thus, the work done during the isothermal compression of the gas is approximately:
\[ \boxed{313.39 , \text{kJ}} \]